Eigenvalue pinching conjecture near the sphere under a lower Ricci bound

Let $n\geq 2$, and let $\epsilon>0$ and $K>0$. For an $n$-dimensional closed Riemannian manifold $(M,g)$, write $\lambda_{n+1}(\bar{\Delta}^{E},M)$ for the $(n+1)$-st eigenvalue of the Einstein operator and $d_{GH}$ for the Gromov–Hausdorff distance. Eigenvalue pinching conjecture. There exists a positive constant $\delta(n,K,\epsilon)>0$ such that if

$$\operatorname{Ric}_g\geq -Kg$$

and

$$d_{GH}(M,S^n)\leq\delta,$$

then

$$\lambda_{n+1}(\bar{\Delta}^{E},M)\leq\epsilon.$$

This asks whether the upper Ricci-curvature bound in the preceding proposition can be removed while retaining eigenvalue pinching for manifolds sufficiently close to the standard sphere. The statement is presented as a conjecture in the source, and its resolution is not established there.